Actually Stellar doesn't choose a path. You (or the software you use) are supposed to define one.
You can actually check various orderbooks and compute the cost to find the best path. You find the conversion rate by finding which offer your operation would match. You can derive the fee by comparing how much the destination asset will cost you against its normal market price.
As the orderbooks update every five seconds, the information can become quickly outdated. Ideally we would have a library/interface that refresh accordingly. As you can see that's quite something to implelent so if your question was "can I do that easily?" well then no not really.
I think the Stellar team mentioned working on setting like this.
As far as I can tell right now the best route is always going through XLM as those are the pairs with more volume by far.